This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role and influence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.
"Local Sub-Gaussian Estimates on Graphs: The Strongly Recurrent Case." Electron. J. Probab. 6 1 - 33, 2001. https://doi.org/10.1214/EJP.v6-95